In many fields of application it is of interest to study the interactions between time series of random variables constrained to be positive, for instance with high-frequency financial variances, durations and trading volumes.
A class of models which is particularly suited to non-negative time series are the Multiplicative Error Models (MEMs) introduced by [1]} to overcome some drawbacks associated with the conventional workarounds used to model non-negative data such as ignoring the non-negativity constraint or taking the logs of the observations. The advantages of modelling a non-negative process using distributions with support on the non-negative orthant are well described by [2]} and can be summarized in these two stylized facts: these distributions do not require complex constraints on their moments to reduce the probability of obtaining negative values during sampling processes and they handle naturally exact zeros (i.e. without adding small constants before taking the logs, that could affect considerably the outcomes).
Specifications of MEMs ask for specific forms of the dynamics in the conditional mean and for the distribution of the multiplicative error term. Diverse choices lead to models of different generality: a scale-factor structure of the conditional mean and a Gamma-distributed error term reduce MEM to the non-negative GARCH model of [1]}, whilst an autoregressive structure on the conditional expectation and a Weibull error term lead to the ACD model of [2]}. The univariate setting has been extended by [5]} to multivariate time series, with the purpose of analysing the interdependence across volatility measures. In the multivariate version of MEM, vector MEM or vMEM, the vector of observations at a given time is represented as the element-by-element product of a conditional mean and a random innovation, with some assumed multidimensional distribution for the innovation vector and a specified dynamics for the conditional mean vector.
[5]} initially proposed a natural extension of the univariate case, assuming that the components of the innovation vector are independently distributed with Gamma densities with equal shape and rate. The same approach has been followed by [7]} and [8]} to analyse the spillover effect between different volatility proxies. To weaken the assumption of stochastic independence among the components of the innovation vector, [9]} explored the possibility to use multivariate Gamma distributions defined on the positive orthant [10]}, and copula-related multivariate distributions with Gamma marginals. We propose a very general and flexible Bayesian semiparametric model specification of the multivariate MEM, with Dirichlet Process Mixture (DPM) innovation errors.
